Soft Matter
HIGHLIGHT
Buckled colloidal monolayers connect geometric
frustration in soft and hard matter
Cite this: Soft Matter, 2013, 9, 6565
Yair Shokef,*a Yilong Han,b Anton Souslov,c A. G. Yodhd and Tom C. Lubenskyd
Buckled monolayers of diameter-tunable microgel spheres constitute a soft-matter model system for
studying geometric frustration in hard-condensed-matter antiferromagnetic materials. In the plane, the
spheres self-assemble to form a triangular lattice. By considering the free volume available to two
spheres slightly out of the plane, one finds an effective antiferromagnetic interaction; each pair of
Received 8th January 2013
Accepted 20th February 2013
neighboring spheres prefers to be one up and one down. However, the topology of the triangular
lattice prevents all pairs from simultaneously satisfying this rule. The micrometer length scale of the
spheres enables direct visualization of the ‘spin’ dynamics at the single-particle level. These dynamics
DOI: 10.1039/c3sm00069a
exhibit glassiness, which originates from the in-plane lattice distortions that partially relieve frustration
www.rsc.org/softmatter
and produce ground states with zigzagging stripes.
I
Geometric frustration
The close-packed triangular lattice constitutes the densest
conguration of monodisperse disks on the plane. This packing
maximizes density both globally and locally, i.e., the most
compact way to position neighboring disks around any given
a
School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail:
[email protected]; Web: http://shokef.tau.ac.il
disk denes a hexagon that belongs to the globally preferred
triangular lattice, see Fig. 1a. On the other hand, packing
spheres in three-dimensional space is geometrically frustrating.1 Locally, the optimal packing contains tetrahedral
motifs of close-packed spheres.2 However, tetrahedra do not ll
space optimally,3,4 and a perfect packing of tetrahedra cannot be
extended to very large structures. Instead, the highest overall
density is obtained by the face-centered cubic (fcc) lattice, which
does not have local tetrahedral motifs and therefore maximizes
b
Department of Physics, Hong Kong University of Science and Technology, Clear Water
Bay, Hong Kong
c
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
d
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA
19104, USA
Yair Shokef completed his PhD in
physics at the Technion – Israel
Institute of Technology. During
2006–2008 he was a postdoctoral
fellow at the University of Pennsylvania, where he focused on
jamming and frustration in
colloidal systems. During 2008–
2011 he was a postdoctoral
fellow at the Weizmann Institute
of Science, where he deepened his
interest in biophysics. Since 2011
he has been a faculty member at
Tel-Aviv University. His research focuses on the non-equilibrium
statistical mechanics of stuck and active so-matter systems,
ranging from frustration and jamming in colloidal systems and
glasses to nonlinear elasticity and biologically driven uctuations
in living cells.
This journal is ª The Royal Society of Chemistry 2013
Fig. 1 Experimental setup. (a) The in-plane packing of a monolayer of spheres is
a triangular lattice. (b) Out-of-plane, neighboring spheres have a tendency to
move away from one another towards opposite walls. (c) Microgel spheres are
composed of cross-linked NIPA polymers. When heated, the spheres shrink
(deswell), and water is squeezed out such that the sample volume fraction is
reduced, yet the spheres remain density-matched with the surrounding fluid.
Soft Matter, 2013, 9, 6565–6570 | 6565
Soft Matter
density globally but not locally. Such frustration is responsible
for the disorder and glassy dynamics5 in a variety of concentrated so-matter systems such as colloidal suspensions,
emulsions, foams and granular matter.6
Interestingly, in many hard-condensed-matter systems
ranging from ice7 to antiferromagnets8–10 and high-Tc superconductors,11 a related inability to satisfy all local constraints
due to geometric incompatibility also arises. The classic
example of such geometric frustration is the antiferromagnetic
Ising model,12 the energy of which is minimized when each
nearest-neighbor pair of spins adopts anti-parallel alignment.
On a square lattice, this structure is easily achieved by the
checkerboard arrangement of ‘up’ and ‘down’ spins shown in
Fig. 2a. The triangular lattice, however, consists of three-particle
plaquettes like the one highlighted in Fig. 2b, in which all
neighboring spins cannot be mutually anti-parallel. In the
ground state, the spins on two bonds of each triangular plaquette are anti-parallel and the spins on the third bond, which
is frustrated, are parallel. Here the desired local order is that of
two anti-parallel neighboring spins. When going to a slightly
larger object of merely three particles, one cannot satisfy all the
local two-particle rules. Although this famous model was the
very rst prototype of geometric frustration, it has rarely been
achieved in real magnetic systems. In the last few years,
however, signicant progress has been made in fabricating
“articial” magnetic systems made of macroscopic building
blocks that interact in a way that mimics the frustration found
in the Ising model.13–20
In this paper, we highlight our recent efforts to study
geometric frustration in so matter, specically in a novel,
tunable colloidal model system consisting of a buckled monolayer of microgel spheres.21 This so-matter system, which
mimics the triangular-lattice antiferromagnetic Ising model,
enabled rst studies of the single-particle dynamics originating
from an analog of spin frustration in hard-condensed-matter
systems. It thus serves as an important tool for advancing the
understanding about frustration effects in both the so- and
hard-matter realms. We employed free-volume calculations to
show how sphere-packing considerations can generate antiferromagnetic interactions in an effective Ising-model description
of our colloidal system.22 However, since the lattice is not rigid
in our so system, mapping to the Ising model on a triangular
lattice is not exact. Positional degrees of freedom permit the
colloidal monolayer to partially relieve geometric frustration
Fig. 2 Antiferromagnetic Ising model: (a) on the square lattice all spins can be
mutually anti-parallel. (b) On the triangular lattice at least one pair of spins in each
triangular plaquette must be parallel.
6566 | Soft Matter, 2013, 9, 6565–6570
Highlight
through local deformations of the lattice,21,22 which, in turn, are
responsible for the structural and dynamical differences
between our observations and the Ising-model predictions.
Furthermore, frustration is oen related to glassiness, and we
have observed slow dynamics21 and jamming,22 which we can
explain by mapping our experimental system onto an Ising
model coupled to an elastic network of springs.23 With this
magneto-elastic interaction present, we exactly calculated24 the
minute differences in entropy between competing ground state
congurations, which are a dening characteristic of various
glassy systems.
II
Experiment
Buckled monolayers of colloidal spheres mimic the triangularlattice antiferromagnetic Ising model.25,26 We conned
micrometer-sized colloidal spheres between parallel plates with
about 1.5-sphere-diameter separation. At high enough densities
the spheres self-assembled into a triangular lattice with buckled
up or down states (Fig. 1) that are analogous to the up and down
spin states of the Ising model. Using this analogy, we will refer
to the up and down congurations of the colloidal particles as
spin congurations. Neighboring spheres prefer opposite states
(Fig. 1b) in order to have more free volume to move, i.e., more
entropy and less free energy. Thus they can be viewed as Ising
spins with effective antiferromagnetic interactions. The analogy
between buckled colloidal monolayers and Ising spins was rst
proposed in the early 1980's,25,26 but at that time the experimentally observed crystalline domains were too small for
meaningful measurements because wall spacing could not be
made uniform enough. Most importantly, image processing
techniques were not well developed, and therefore it was difficult to carry out quantitative measurements of these systems. In
the work highlighted here, we generated large crystalline
domains by using novel thermal-sensitive N-isopropyl acrylamide (NIPA) microgel spheres.27 These spheres exhibit shortrange repulsion,28 and their hydrodynamic diameter decreases
roughly linearly from 850 nm at T ¼ 20 C to 650 nm at T ¼
30 C,29 because NIPA polymers become less hydrophilic at
higher temperatures (Fig. 1c). Thus, crucially, the ability to
temperature-tune the particle diameter and sample packing
fraction enabled us to vary the strength of the effective antiferromagnetic interaction between particles on the triangular
lattice. Finally, since most of their volume is water, the spheres
remain density-matched with the surrounding uid, and gravitational effects are negligible.
Fig. 3a shows the monolayer at high temperature. The
spheres self-assemble to form an in-plane triangular lattice. We
focus our microscope image plane to a position in the top half
of the experimental cell; in this case, spheres that are close to
the top wall appear bright, whereas those close to the bottom
wall are out of focus and appear darker. We measured a clear
bimodal brightness distribution, indicating that spheres have a
strong tendency to be close to either wall, rather than to stay in
the center of the cell. Furthermore, we observed antiferromagnetic behavior, namely a statistical preference for neighboring
spheres to be close to opposite walls.21 As temperature is
This journal is ª The Royal Society of Chemistry 2013
Highlight
Soft Matter
bonds tend to arrange into parallel zigzagging stripes, as shown
in Fig. 3b, instead of the more random distributions, which are
preferred in the Ising model. We found that this discrepancy is
due to the ability of the lattice to deform. This, in turn, can
relieve some frustration and reduce the ground-state degenpffiffiffi
eracy from eN in the Ising model to e N here, i.e., the zerotemperature entropy is reduced from extensive (S0 N) to subpffiffiffiffi
extensive (S0 N ), where N denotes the number of particles in
the system.
Video microscopy of these micron-size colloidal particles
enables us to identify the ‘spin state’ of each particle and also to
track the Brownian dynamics of individual particles by image
processing. Representative movies of the sample dynamics are
available online in the ESI of ref. 21. As temperature is lowered
these dynamics become slower; here we quantify the dynamics
by the time autocorrelation function of the ‘spin state’ shown in
Fig. 3c. These autocorrelation functions exhibit a stretched
exponential decay, which can be viewed as a superposition of
several exponential decays with different relaxation times. The
overall relaxation time dramatically increases over a small
temperature range.
III
Isosceles tiling
We can understand why the observed zigzagging-stripe congurations are selected out of the multitude of disordered Ising
ground-state congurations from the following packing arguments. At low temperature, the microgel spheres swell, and the
system seeks congurations that enable close-packing of the
colloidal spheres. For particles that are roughly hard spheres,
the closest distance that two neighboring spheres can approach
one another is equal to the sphere diameter d. For a frustrated
bond (i.e., two neighboring spheres close to the same wall), the
vector connecting the sphere centers is in the plane of
connement. However, for a bond that satises the effective
antiferromagnetic interaction (i.e., neighbors wherein one
Fig. 3 Peculiar antiferromagnetic behavior. (a) At T ¼ 27.1 C, the spheres form an
in-plane triangular lattice, with roughly half of the spheres up (bright) and half down
(dark), and with a statistical preference for neighboring spheres to be at opposite
heights, as in an antiferromagnet. (b) At T ¼ 24.7 C, the frustrated bonds form
predominantly zigzagging stripes, which represent only a small fraction of the possible
Ising model ground state. (c) The two-time correlation function C(t) ¼ hzi(0)zi(t)i
between the single-particle spin state, zi(t) ¼ 1, decays as a stretched exponential of
time, C(t) ¼ exp[(t/s)b] with a relaxation time s, which dramatically increases with
decreasing temperature, and a stretching exponent b, which decreases toward zero as
the temperature is decreased, see the inset. Figure adapted from ref. 21.
lowered, the sphere diameter increases, and the geometric,
effective antiferromagnetic interaction between particles
becomes stronger.22 This effect leads to an increase in the
measured probability of nding anti-parallel neighboring
‘spins’. Despite this behavior, which generally agrees with the
Ising model, we observed that at low temperature the frustrated
This journal is ª The Royal Society of Chemistry 2013
Fig. 4 Isosceles tiling: (a) the in-plane distance between a frustrated pair is the
particle diameter d, while between a pair that manages to satisfy the effective
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
antiferromagnetic interaction it is x ¼ d2 ðh dÞ2\d. (b) For this reason, each
triangular plaquette in the Ising ground state deforms to an isosceles triangle
(in-plane) with two short edges x and one long edge d. (c) Of the configurations
of a particle and its six neighbors that are allowed in the Ising ground state, the
corresponding isosceles triangles tile the plane only when the central particle has
exactly two frustrated neighbors. Up particles are colored yellow and down
particles are red, and in each isosceles triangle, the head angle is marked in
purple. Figure adapted from ref. 21.
Soft Matter, 2013, 9, 6565–6570 | 6567
Soft Matter
sphere is close to the top wall, and one close to the bottom wall),
this vector is tilted with respect to the horizontal direction, and
its projection onto the plane of connement gives a shorter inplane distance, x < d, see Fig. 4a.
As a result, in the ground state of the antiferromagnetic Ising
model, each triangular plaquette has one frustrated bond and
will deform to an isosceles triangle with two bonds of length x
and one bond of length d, see Fig. 4b. The close-packed state
with the maximal density is then obtained by tiling the plane
with such isosceles triangles. To see which spin states are
compatible with this tiling requirement, we now consider
possible congurations of a sphere and its six neighbors, which
also denes six triangles around the central sphere, see Fig. 4c.
We rst require that each triangle contain only one frustrated
bond. Up to rotations and spin inversions, the Ising-groundstate rule is satised in the ve congurations shown in Fig. 4c.
Of these ve congurations, only two correspond to a fully
space-lling tiling by isosceles triangular plaquettes. These two
local states, selected by packing considerations, generate
congurations of zigzagging stripes. Thus, each particle has two
frustrated neighbors, and at each row the parallel stripes may
continue on a straight line, or turn and bend, as shown experimentally in Fig. 1b and theoretically in Fig. 6b and c.
This structural difference between our colloidal antiferromagnet and the Ising model is also responsible for the different
dynamical behavior we observe. In the Ising model, the ground
state is not only disordered and highly degenerate, it also has
high connectivity, i.e., the system can evolve from one groundstate conguration to another by transitions that do not require
any excitation energy and, thus, occur easily at arbitrarily low
temperatures. This phenomenon is seen most clearly by the
local zero-energy mode that the rightmost conguration in
Fig. 4c represents. Here, the central particle has three frustrated
bonds and three satised bonds, and thus has the same energy
regardless of the spin state of the central particle. Flipping the
central spin costs no energy and enables the system to more
easily access all ground-state congurations.30 It is also
responsible for the rapid equilibration of the Ising model at any
temperature. For our colloidal antiferromagnet, however, the
local spin ip costs free-energy. Having no venue to relax, the
system may get stuck in a metastable state, either relaxing very
slowly,21 or sometimes never reaching equilibrium.22
Highlight
nite temperature), the thermodynamically stable state is the
one with the largest free volume, and it is not clear which
conguration has minimum free energy. This situation is
similar to that which arises when packing spheres in three
dimensions wherein the same maximal density may be
obtained by stacking hexagonally packed layers, with any arbitrary sequence of lateral shis, leading in the two limiting
sequences to the fcc lattice and to hexagonal close packing
(hcp). In this case, minute entropic differences are believed to
select fcc over hcp,37–46 but exact results are still lacking.
Before analyzing the entropic difference between straight
and bent stripes, we consider the generic free-energy landscape
of a glassy system. Typically, we expect to nd in this landscape
many local minima in which the system may be trapped and be
metastable, plus a single (or slightly degenerate) minimum
which is deeper than the others and is the equilibrium state of
the system, see Fig. 5a. When rapidly cooled, the system oen
will get stuck in one of the metastable states; only if it is cooled
extremely slowly, can the system reach the true equilibrium
state. In our colloidal antiferromagnet, the free-energy landscape has a highly degenerate ground state, namely many
minima (all at the same free energy), and what makes one
conguration more stable thermodynamically than the others is
the width, i.e., the entropy associated with it, see Fig. 5b.
To calculate the free-volume contribution to the entropy of
each zigzagging-stripe conguration, we approximate our
colloidal system with an Ising model on an elastic network.23,24
Namely, on top of the discrete, spin degrees of freedom, we now
allow each spin to have a continuous displacement in the plane,
see Fig. 5c. The total energy of the system is comprised of the
antiferromagnetic interaction between neighboring pairs, the
strength of which now decays linearly with the separation
between them, together with an elastic energy representing the
stretching or compression of the identical springs connecting
IV Order by disorder
Even though a frustrated system can have a highly degenerate
ground state at zero temperature, at nite temperature entropy
becomes relevant, and the free energy of hard-sphere-like
systems is minimized by selecting the state with the greatest
entropy. Thus, thermal uctuations can induce order due to
small uctuations that are energetically soer than the excitations associated with the disordered ground-state congurations, see e.g., ref. 9 and 31–36. Our tiling scheme proves that all
possible realizations of zigzagging stripes have the same
maximal density, which is the analog of the zero-temperature
ground state. However, when the packing fraction is somewhat
smaller than this maximal value (this corresponds to being at
6568 | Soft Matter, 2013, 9, 6565–6570
Fig. 5 Order by disorder: schematic representation of energy landscape in (a)
glass, and (b) deformable Ising model. Red circles mark the lowest free-energy
state, which the system prefers in equilibrium, i.e., the lower energy state in (a),
and the highest entropy state in (b). (c) Local configurations corresponding to
straight and zigzag stripes in the elastic Ising model. Yellow and red circles
represent up and down spins.
This journal is ª The Royal Society of Chemistry 2013
Highlight
each pair of neighboring spins. Our treatment is related to
previous work on magneto-elastic coupling in this context,47,48
but it is more precise since we allow arbitrary deformations of
the lattice.
For the deformable antiferromagnetic Ising model, the width
of each free-energy minimum is related to the phonon spectrum
around the corresponding deformed conguration, and we can
thus exactly calculate the low-temperature entropic contribution
of the phonons to the free energy. Interestingly, we found that for
all values of the magneto-elastic coupling, straight stripes are
entropically favored over zigzags.24 This is consistent with theoretical arguments showing that generally the most anisotropic
ground state conguration has the highest entropy.33 We veried
this result with numerical simulations in which our elastic Ising
model was cooled at different rates. Fig. 6 shows that when
rapidly cooled (panel a), only partial antiferromagnetic order is
obtained, as can be seen from the fact that there are triangular
plaquettes with all three particles in the same Ising state. For
moderate cooling rate (panel b), the system reaches a conguration that belongs to the ground state of the Ising model (each
plaquette contains only one frustrated pair), which has the
Soft Matter
aforementioned zigzagging-stripe structure. As the cooling rate is
decreased even more (panel c), the persistence length of the
stripes grows and they become closer to the straight stripes which
constitute the equilibrium state we have predicted.
Although the equilibrium state is that of straight stripes, for
virtually any preparation history the system gets jammed in a
zigzagging stripe conguration. Such congurations are metastable only in a very weak sense, however, they have the same
maximal packing fraction (corresponding to the energy in the
Ising model) as straight-stripe congurations, and the only
thermodynamic preference of straight stripes over zigzags is
that straight stripes have a very small entropic advantage.
However, passing between different close-packed states
pffiffiffiffi
requires collective rearrangements involving the order of N
particles, and requiring additional free-volume, which is not
necessarily available for the system.
V
Conclusions and outlook
In this paper, we have described our experimental and theoretical
efforts to study a novel so-matter system, consisting of diameter-tunable microgel spheres. When positioned between parallel
plates that are separated by about 1.5 sphere diameters, the
spheres self-assemble to form a buckled triangular-lattice
monolayer. Packing considerations in the buckling of this
monolayer generate effective interactions and frustration that are
similar to those found in the antiferromagnetic Ising model on
the triangular lattice. However, motion of the spheres in the
plane of connement partially relieves the frustration, giving rise
to glassy dynamics and to partial order in the form of zigzagging
stripes. Current techniques enable experimental tracking of the
dynamics of individual particles in this so-matter system, and
thereby to learn about related frustration effects and frustration
relaxation dynamics of hard-condensed matter magnetic materials. Moreover, we expect that the visualization of the singleparticle dynamics in our colloidal system will directly advance the
theoretical understanding of the dynamical properties of various
currently studied mesoscopic frustrated systems.13–20
Future work on our colloidal antiferromagnet should focus on
analyzing its non-equilibrium dynamics. For example, by
manipulating individual particles with laser tweezers, one could
measure spatio-temporal correlations between different particles
in the system, and use this information to learn about relaxation
processes in glassy materials. Furthermore, with holographic
optical tweezers,49–51 one could bring the system to an arbitrary
conguration, and monitor the relaxation processes that will
occur aer the trapping potential is turned off. Finally, it would
be interesting to relate the packing frustration investigated here
to that found in other colloidal systems such as self-assembled
Janus particles,52,53 colloidal spheres packed in narrow tubes,54–57
and to the aforementioned fcc vs. hcp question.40
Acknowledgements
Fig. 6 Final configurations of our elastic Ising model for different cooling rates, r.
Yellow and red circles represent up and down spins. Blue lines guide the eye to
identify the zigzagging stripe patterns. Figure reproduced from ref. 24.
This journal is ª The Royal Society of Chemistry 2013
We thank Ahmed Alsayed, Chris Henley, Valery Ilyin, Randy
Kamien, Andrea Liu, Itamar Procaccia, Ido Regev, and Peter
Yunker for helpful discussions and technical assistance. This
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work was supported by National Science Foundation grants
DMR12-05463, PENN-MRSEC-DMR11-20901, by NASA grant
NNX08AO0G, and by Israel Science Foundation grants 617/12,
1730/12.
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